Emerald Dragon and Private eye dol both have a memory-backup function. The odd part is, the two are entirely incompatible with eachother -- system memory backups you make in Emerald Dragon can't be copied to and from the system in Private eye dol, and vice versa. You're pretty much stuck with one or the other...
Emerald Dragon - You hold a button (or is it a d-pad direction?) when starting the game to boot in to the MB129 menu. The game doesn't support saving to the MB128 for some stupid reason, but you can copy memory blocks back and forth between the system and backups you make on the MB128. You cannot see files saved directly to the MB128 here, or Private eye dol memory backups, only ED's memory backups. In Private eye dol, you see the sizes of these files on the MB128, but not what's in them. The size is the total of all files in that block.
Private eye dol - This has a more comprehensive save backup program accessible from the menu. You can make memory backups as in ED, though they're incompatible of course, and also view files on the system and MB128, with file sizes visible. Private eye dol memory backups are all the maximum size of 2048, so they don't change in size based on the size actually used, unlike Emerald Dragon.
I haven't used the Private eye dol memory manager recently, I think it can do a few more things but had some limitations as well. Maybe I'll go back to it and check that out.
As for the MB128 itself, I find it useful to use batteries wit ha tester, so I can check them every once in a while. That onboard capacitor will not hold a charge for long, so if the batteries run out you probably will have lost your saves before you notice. A set of good batteries should last 6+ months, though, so they do last a while. And it DOES hold saves while the system is on (I presume it gets power from the system), so you can remove the battery box without worry, though that capacitor will hold them for a little while while you switch out batteries of course.